We believe that the threshold value is a function of remaining position size such that it increases as the position size decreases and converges to \(\operatorname{\textup {VaR}}_q[C_\infty ]\)  as the position vanishes.
Moreover, the change in the aggressiveness around the threshold also depends on the remaining position size.
To formalize this behavior, one may adopt an alternative formulation of the problem with an extra state variable representing the cost realized so far (i.e., a Markov policy defined on the augmented state space \((X_t, C_t)\) ), which is in fact the approach suggested by [1]} for a general class of control problems with a CVaR objective.
This might be a topic of future research.
